8. Hypotheses 8.2 The Likelihood ratio at work
K. Desch – Statistical methods of data analysis SS10
data
data
data
n(s b)
nsdatan
(b)
data
(s b)e
n ! sQ e (1 )
bbe
n !
8. Hypotheses 8.2 The Likelihood ratio at work
K. Desch – Statistical methods of data analysis SS10
in general, use MCto caluclate thep.d.f. for Qunder the s+band b hypotheses
8. Hypotheses 8.2 The Likelihood ratio at work
K. Desch – Statistical methods of data analysis SS10
Alex Read, "Modified Frequentist Analysis of Search Results (The CLs Method)" CERN 2000-205 (30 May 2000)
http://cdsweb.cern.ch/record/451614/files/open-2000-205.pdf
8. Hypotheses 8.2 The Likelihood ratio at work
K. Desch – Statistical methods of data analysis SS10
Combination of several channels is easy:
For Poissonian p.d.f.s
tot ichan
Q Q
data
data
n(s b)
pois data datatot n
(b)chan chanpois data
data
(s b)e
P (n ;s b) n !Q
P (n ;b) be
n !
i i itot data,i i i i tot data,i
chan chani i
s b slnQ n ln( ) (s b b ) s n ln(1 )
b b
a channel contributes according to ln(1+s/b) to the log LHR!
8. Hypotheses 8.2 The Likelihood ratio at work
K. Desch – Statistical methods of data analysis SS10
Example:
„Channel“ can mean:- different experiments- different search topologies (event types) in a single experiment- different bins (e.g. reconstructed mass, discr. variable) in the same topology with different s/b
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
In fall 2000 the 4 LEP experiments released their (perliminary) resultsin the search of the Higgs boson.
This was a search of 4 different experiments (detectors) (ALEPH, DELPHI, L3, OPAL) each in 4 different search topologieswith small numbers of background and small numbers of expectedsignal events (in case there was a Higgs boson)
No single experiment had enough sensitivity to discover a Higgs atmH >~ 110 GeV
Combination was done using the likelihood ratio as a test statistic
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Production:
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Decay:
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Search topologies:
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
For each selected candidate event two main discriminating variableswere used:
b-quark content of the jets(Higgs decays to b-bbar)
reconstructed mass of thetwo jets which are assignedto the Higgs candidate(signal peaks at Higgs mass, background peaks at Z mass)
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
For each bin in (mrec, b-tag) the expected s/b was calculated from MC simulation (for a range of hypothetical true Higgs masses)
For each hypothetical true Higgs mass („test mass“) a LHR test wasperformed
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
simplest combination
throw all eggs in one basketthen LHR-Test for 1 channel
sensitivity not optimal
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Full LHR test statistic for each experiment (for a single test mass m=114 GeV)
full black line:observed -2 lnQ
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Full LHR test statistic for each search topology(for a single test mass m=114 GeV)
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Discovery ? (1- CLb) < Prob(n * )
„100*(1-CLb)“ percent of experiments would yield a test statistic -2lnQ smaller than the observed one, if no signal would be present“
8. Hypotheses 8.3 A real example: Higgs search at LEP
K. Desch – Statistical methods of data analysis SS10
Exclusion ? CLs (mtest) < 5% „95% CL exclusion“
„Only 100*CLs percent of experiments would yield a -2lnQ larger (i.e. more BG-like) than the observed one if a signal (at the test mass) would be present